In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If L n be the number of the inner boundary points of random walk range in the n steps, we prove lim n→∞ Ln n exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on two dimensional square lattice is of the same order as n (log n) 2 .
This paper considers the question: how many times does a simple random walk revisit the most frequently visited site among the inner boundary points? It is known that in Z 2 , the number of visits to the most frequently visited site among all of the points of the random walk range up to time n is asymptotic to π −1 (log n) 2 , while in Z d (d ≥ 3), it is of order log n. We prove that the corresponding number for the inner boundary is asymptotic to β d log n for any d ≥ 2, where β d is a certain constant having a simple probabilistic expression.
As Dembo (2003, 2006) suggested, we consider the problem of late points for a simple random walk in two dimensions. It has been shown that the exponents for the number of pairs of late points coincide with those of favorite points and high points in the Gaussian free field, whose exact values are known. We determine the exponents for the number of j-tuples of late points on average.
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